Approximate Degree Lower Bounds for Oracle Identification Problems.

The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string $x \in \{0, 1\}^n$ given possibly non-standard oracle access to it. We apply this framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $\Omega(n/\log^2 n)$ for each.